Deformation theory and Lie algebras

3 An example

Here we give an example of what Costello-Gwilliam call an elliptic formal moduli problem. This is a sheaf of formal moduli problems on a manifold such that the associated Lie algebra can be locally presented using a complex of elliptic operators. This example models the free scalar field theory.

Consider a Riemannian manifold \(M\), and consider the complex \(0 \to C^\infty ( M)\xrightarrow {\Delta }C^\infty (M)\), where \(0\) is in degree \(0\), and \(\Delta \) is the Laplacian. We make this a dgla by giving it trivial Lie bracket. We claim that this corresponds to the formal moduli problem classifying deformations of the function \(0\) as a harmonic function. To see this, we will explicitly describe and interpret \(\MC _0(\mg \otimes m)\) and \(\MC _1(\mg \otimes m)\) for \(m\) the maximal ideal of an Artinian local cdga \(R\).

Because the Lie bracket is trivial, \(\MC _0(g\otimes m)\) is the kernel of the differential in degree \(-1\), i.e the map

\[C^{\infty }(M) \otimes m_1 \oplus C^{\infty }(M)\otimes m_0 \xrightarrow {(-d_R,\Delta )} C^{\infty }(M)\otimes m_0\]

In otherwords, we have a function \(f \in C^*(M)\otimes R_0\) that vanishes modulo the maximal ideal (i.e a deformation of the zero function) and a homotopy \(g \in C^*(M)\otimes R_1\) satisfying \(\Delta f= d_Rg\) (i.e a witness to \(\Delta f\) being nulhomotopic).

Next, we consider \(\MC _1(g\otimes m)\). Here there are four pieces of data, which must satisfy three equations.

\[ f(t) \in C^{\infty }(M)\otimes m_0[t]\]

\[ g(t) \in C^{\infty }(M)\otimes m_1[t]\]

\[ h_1 \in C^{\infty }(M)\otimes m_1[t]dt\]

\[ h_2 \in C^{\infty }(M)\otimes m_2[t]dt\]

The first equation these satisfy is \(\Delta f(t) = d_Rg(t)\). This is interpreted as above, meaning \(f(t)\) is a family of deformations and \(g(t)\) is a family of nulhomotopies of their Laplacians.

The second equation they satisfy is \(d_Rh_1 = \frac {df}{dt}dt\). In otherwords, the homology class of \(f\) remains constant as a function of \(t\), as witnessed by \(h_1\).

To understand the last equation, we first note that because \(\Delta ,\frac d {dt}dt, d_R\) all commute up to signs, the first two equations imply that \(d_R(\frac {dg}{dt}dt + \Delta h_1) = 0\). \(h_2\) is a witness of the fact that \(\frac {dg}{dt}dt + \Delta h_1\) is actually exact: the last equation is \(\frac {dg}{dt}dt + \Delta h_1 = d_Rh_2\).

In general, \(\MC _n(g\otimes m)\) consists of higher homotopies and higher coherence data of deformations.